For EdgeWeight->Automatic the weight of each edge is set to the number of pixels in the corresponding skeleton connection. If more than one connection exists, the shortest one is chosen to represent the weight of the edge.

However, looking at the AdjacencyGraph, it seems to always fill in 0's and 1's, so I assume this not to be working.

1 Answer
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This is more of a remark than a solution, but it was too long for a comment.

First of all, although it's not visible in the plot, MorphologicalGraph with EdgeWeight -> Automatic does set edge weights for the edges. However, these are not based on the thickness of the edges but on their length. For example, for the example in the original post the edge weights of g are equal to

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